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Highest weight modules and \(b\)-functions of semi-invariants. (English) Zbl 0821.22006
The author gave a criterion for irreducibility of generalized Verma modules in a preceding paper, where the author clarified that the irreducibility is closely connected with the roots of \(b\)-functions of semi-invariants. Thus, in order to judge when the generalized Verma module is irreducible, it is necessary to calculate the \(b\)-functions of semi-invariants. In this paper, the author develops techniques to compute \(b\)-functions and formulates a new conjecture which would eliminate a restriction of the criterion given in the preceding paper. This paper contains many interesting examples of the calculations of holonomy diagrams. They would give us remarkable information on micro-local structure of semi-invariants.
Reviewer: M.Muro (Yanagido)

MSC:
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
17B35 Universal enveloping (super)algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
14M17 Homogeneous spaces and generalizations
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
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